Describe the type of optimal control problem that is amenable to analysis using Pontryagin's Maximum Principle.

A firm has the right to extract oil from a well over the interval $[0, T]$. The oil can be sold at price $£ p$ per unit. To extract oil at rate $u$ when the remaining quantity of oil in the well is $x$ incurs cost at rate $£ u^{2} / x$. Thus the problem is one of maximizing

$$\int_{0}^{T}\left[p u(t)-\frac{u(t)^{2}}{x(t)}\right] d t$$

subject to $d x(t) / d t=-u(t), u(t) \geqslant 0, x(t) \geqslant 0$. Formulate the Hamiltonian for this problem.

Explain why $\lambda(t)$, the adjoint variable, has a boundary condition $\lambda(T)=0$.

Use Pontryagin's Maximum Principle to show that under optimal control

$$\lambda(t)=p-\frac{1}{1 / p+(T-t) / 4}$$

and

$$\frac{d x(t)}{d t}=-\frac{2 p x(t)}{4+p(T-t)}$$

Find the oil remaining in the well at time $T$, as a function of $x(0), p$, and $T$,